Read the text below first and then

Start the applet

Note: To run the applet in its own window, click on "Start the Applet" with the right mouse button and then select "Open in New Window" from the pop-up menu.

The purpose of this applet is to provide you with a review of very elementary integration techniques. That is, integration based on the chain rule. Nearly all of the 50 problems that occur are of the form

For example

Your task is one of doing integrals like this in your head, without pencil or paper.

You may have learned to do integrals by substitution:

This is a reasonable way when you first come to the subject. However, once you have done 20 or 25 problems by substitution you should move on to the next level, doing them in your mind.

When the applet screen appears, and after you click on the "next problem" button an integral and four possible solutions appear on the screen. Figure out the answer and then click on the solution.

The computer keeps a running score on your answers. You get 3 points if you select the right answer on your first choice and 1 point if you make the right choice on the second selection. If your second choice is wrong the program tells you the answer.

A small number of the harder problems require you first do some simple algebra and/or apply trigonometric identities. calculate, mentally,

For example, to deduce that

you need to figure out that

(sin(x)-cos(x)^2 = 1 -2sin(x)cos(2x) = 1 - sin(2x)

In general, it is assumed that you "know" the following anti-derivatives:

(I) Powers, logarithmic, and exponential integrals:


(II) Trigonometric integrals:



(III) Inverse Trigonometric integrals: